Carrollian Conformal Bases

• Carrollian conformal bases are specialized basis choices on degenerate Carroll manifolds that organize generators, fields, and operators for transparent symmetry action.
• They emerge via ultra-relativistic contraction of the Poincaré algebra and encode infinite extensions like supertranslations, crucial for flat-space holography and null-infinity scattering.
• Applications include geometric formulations, field representations in Carrollian electrodynamics, and intertwiners linking bulk Poincaré modules with boundary Carrollian conformal modules.

Carrollian conformal bases are basis choices adapted to conformal Carroll symmetry on degenerate Carrollian manifolds and, in flat-space holography, on null infinity. In the literature they organize generators, fields, states, or operators so that the conformal Carroll algebra, its infinite supertranslation extension, or boundary versions such as the boundary Carrollian conformal algebra act transparently. In this sense, the subject includes the geometric basis of Carroll structures, scalar/vector field bases obtained by ultra-relativistic contraction, boundary-preserving mode bases, null-infinity operator bases for scattering, and representation-theoretic intertwiners between bulk Poincaré modules and boundary Carrollian conformal modules (Duval et al., 2014, Banerjee et al., 2020, Nguyen et al., 2023, Bagchi et al., 2024, Liu et al., 3 Jun 2026).

1. Geometric and algebraic setting

A Carroll manifold is introduced as a triple

$(C,g,\xi),$

where $C$ is a $(d+1)$-dimensional manifold, $g$ is a symmetric covariant tensor of rank $d$, and $\ker g$ is generated by a nowhere-vanishing complete vector field $\xi$. In flat coordinates $(x^A,s)$, the standard Carroll structure is

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$

The coordinate $s$ is the Carrollian “time” (Duval et al., 2014).

Conformal Carroll transformations of level $C$0 are defined by

$C$1

or infinitesimally

$C$2

The integer $C$3 labels a family of conformal extensions; $C$4 is especially important because it gives the algebra relevant to BMS in the appropriate dimension (Duval et al., 2014).

In later conformal Carroll field-theory constructions, flat Carrollian geometry is written as a degenerate manifold $C$5 with

$C$6

and conformal Carroll isometries are generated by vector fields $C$7 obeying

$C$8

The finite generators are

$C$9

$(d+1)$0

$(d+1)$1

with $(d+1)$2. The infinite extension is the Abelian ideal of supertranslations

$(d+1)$3

for arbitrary spatial function $(d+1)$4, with $(d+1)$5, $(d+1)$6, and $(d+1)$7 embedded as the special choices $(d+1)$8, $(d+1)$9, and $g$0 (Banerjee et al., 2020).

The ultra-relativistic origin of this algebra is central. The Carrollian algebra is obtained from the Poincaré algebra by taking the speed of light to zero, and the conformal version similarly follows. Conformal Carrollian groups are known to be isomorphic to Bondi-Metzner-Sachs groups, so the algebraic basis is also an asymptotic-symmetry basis at null infinity (1901.10147). This suggests that the plural “Carrollian conformal bases” reflects several technically distinct realizations of one underlying symmetry framework.

2. Field representations and Carrollian primary data

A basic field-theoretic basis organizes fields into $g$1 scalars $g$2 and vectors $g$3, with definite scaling dimension $g$4. In the ultra-relativistic contraction of relativistic electrodynamics, the two inequivalent boost representations are the electric and magnetic sectors. They are packaged as

$g$5

with

$g$6

Special conformal transformations and supertranslations act locally on this scalar/vector basis, and the supertranslation ansatz

$g$7

is checked to be self-consistent, with

$g$8

for the conformal Carroll algebra (Banerjee et al., 2020).

A complementary intrinsic construction appears at null infinity. On $g$9, one chooses at the origin a finite-dimensional irreducible representation of the stability subgroup

$d$0

For spin $d$1, the field at the origin is a symmetric traceless $d$2 tensor $d$3, with

$d$4

Away from the origin,

$d$5

and the field transforms as

$d$6

Matching to massless irreducible representations fixes

$d$7

for these Carrollian conformal fields (Nguyen et al., 2023).

The scalar sector is especially important. The electric conformal Carrollian scalar on null infinity has action

$d$8

equation of motion

$d$9

and conformal weight

$\ker g$0

This module is interpreted as the flat-space limit of the singleton representation, and the corresponding flat-space analogue is called the simpleton (Bekaert et al., 2022). In a holographic formulation on $\ker g$1, the same scalar appears with electric and magnetic Carrollian actions that are equivalent, both on-shell and off-shell, up to a non-local inversion of the shifted Laplacian $\ker g$2; the solution space

$\ker g$3

carries a non-unitary indecomposable module of the Carroll, Poincaré, and BMS algebras (Bekaert et al., 2024).

3. Null infinity, holography, and particle/operator bases

The null conformal boundary $\ker g$4 of Minkowski spacetime carries a degenerate Carrollian metric. In retarded coordinates, one has

$\ker g$5

or, on $\ker g$6,

$\ker g$7

The Poincaré group acts on $\ker g$8 as the group of Carrollian conformal isometries, and in the Carrollian realization the quadratic Casimir vanishes identically,

$\ker g$9

so fields on $\xi$0 can only realize massless representations (Nguyen et al., 2023, Kulkarni et al., 8 Aug 2025).

This geometry underlies a boundary basis for scattering. In one formulation, the asymptotic position basis is

$\xi$1

and the corresponding Carrollian fields on null infinity are obtained by an embedding-space projection and a null-infinity limit (Salzer, 2023). In another, local Carrollian primaries $\xi$2 are related to massless creation operators by the Fourier–Mellin map

$\xi$3

The boundary coordinates $\xi$4 are dual to null momentum variables $\xi$5, with $\xi$6 conjugate to $\xi$7 (Nguyen, 13 Nov 2025).

In general dimensions, the boundary operator induced from a bulk free massless scalar is

$\xi$8

and it transforms as a conformal Carrollian primary with

$\xi$9

Its $(x^A,s)$0-descendants $(x^A,s)$1 transform with shifted dimension $(x^A,s)$2 (Kulkarni et al., 8 Aug 2025).

The relation to celestial holography is close but not identical. A sourced conformal Carrollian field theory on $(x^A,s)$3 uses time-dependent Carrollian primaries $(x^A,s)$4, while celestial operators $(x^A,s)$5 are obtained by an integral transform in retarded or advanced time,

$(x^A,s)$6

The Carrollian weights satisfy

$(x^A,s)$7

(Donnay et al., 2022). A notable distinction is that, in the simpleton representation, supertranslations act nilpotently rather than diagonally: $(x^A,s)$8 so there is no ordinary momentum basis for the simpleton (Bekaert et al., 2024).

4. Boundary-adapted and low-dimensional bases

In $(x^A,s)$9 dimensions, the intrinsic Carrollian conformal algebra is generated by local transformations

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$0

with modes

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$1

The mode algebra is

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$2

Within this basis, Carrollian multiplets are defined as indecomposable boost representations, the energy-momentum tensor modes furnish the centrally extended $$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$3D Carrollian conformal algebra, and contour-integral formulas provide an OPE/commutator dictionary without radial quantization (Saha, 2022).

In $$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$4D Carrollian conformal field theories on a null cylinder, the full algebra is generated by

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$5

Placing boundaries at $$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$6 leads to the boundary Carrollian conformal algebra, with boundary-preserving generators

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$7

or equivalently $$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$8. Their algebra is

$$C^{d+1}=\mathbb{R}^d\times \mathbb{R},\qquad g = \delta_{AB}\,dx^A dx^B,\qquad \xi=\partial_s.$$9

$s$0

$s$1

This algebra is presented as distinct from the usual BCFT boundary Virasoro algebra, and it also arises as the constraint algebra of open null strings with Dirichlet boundary conditions (Bagchi et al., 2024).

The BCCA later acquired an algebraically sharper basis. In the original basis it is generated by

$s$2

but the centreless BCCA is filtered but not graded. A new basis

$s$3

rewrites the algebra as

$s$4

with

$s$5

$s$6

This is the paper’s key “Carrollian conformal basis” for the BCCA, because it makes the descending filtration manifest and enables intrinsic Whittaker-module constructions (Buzaglo et al., 29 Aug 2025).

5. Dynamical and interacting realizations

A concrete interacting Carrollian conformal basis was constructed in the magnetic sector of Carrollian electrodynamics. The magnetic ultra-relativistic limit gives

$s$7

but these do not come from a local action and fail the Helmholtz conditions. A minimal set of new fields

$s$8

is therefore added. After imposing Helmholtz integrability and conformal Carroll invariance, the final equations are

$s$9

$C$00

$C$01

with free Lagrangian

$C$02

The Hessian is invertible, the theory is free of constraints and gauge redundancies, and quartic interactions are fixed by symmetry to

$C$03

The strong dynamical invariance under $C$04, $C$05, and $C$06 is verified, and the Noether charge algebra reproduces the conformal Carroll algebra exactly, with no central extension (Banerjee et al., 2020).

A broader on-shell perspective comes from ultra-relativistic limits of relativistic conformal theories. Explicit examples include Carrollian scalars, fermions, electromagnetism, Yang-Mills theory, and general gauge theories coupled to matter fields. Concentrating on the equations of motion, it is shown that even in dimensions $C$07, there is an infinite enhancement of the underlying symmetry structure, generated by supertranslations $C$08 (1901.10147).

The same algebra has also been realized dynamically through deformed light-cone null reduction. For a free massless complex scalar in a deformed light-cone background, null reduction along $C$09, followed by $C$10, rescaling, and the Carroll limit $C$11, gives

$C$12

From stress-tensor formulas one obtains the dynamical basis

$C$13

whose commutators reproduce the known kinematic Carrollian conformal algebra (Saha et al., 8 Oct 2025).

For a complex vector field in a deformed light-cone background, the null-reduced Carrollian theory becomes

$C$14

The $C$15 null component and the transverse spatial components survive as decoupled complex scalar fields, while the $C$16 null-direction component vanishes. The preferred generator basis is

$C$17

and the secondary constraint $C$18 is essential for deriving the correct spatial translations (Zeng, 6 Feb 2026).

6. Representation-theoretic extensions

The conformal Carrollian scalar has a distinguished representation-theoretic status. The on-shell electric conformal Carrollian scalar on null infinity can be interpreted as the flat-space limit of the singleton representation of the conformal algebra, while the corresponding flat-space higher-spin algebra is

$C$19

with $C$20 the annihilator ideal of the simpleton module. The full symmetry algebra of the kinetic operator $C$21 is larger, yielding extended BMS and higher-spin BMS-type structures (Bekaert et al., 2022).

Supersymmetric extensions display a similar pattern of basis enlargement. In $C$22 and $C$23, nontrivial Carrollian superconformal algebras are isomorphic to super-Poincaré algebra of $C$24 and $C$25, respectively, and neither construction requires R-symmetry to ensure algebraic closure. The same framework also produces a singlet super-BMS$C$26 algebra and two multiplet chiral super-BMS$C$27 algebras (Zheng et al., 28 Mar 2025).

A recent representation-theoretic development makes the notion of Carrollian conformal basis fully explicit as an intertwiner problem. The Poincaré–Carrollian intertwiner

$C$28

maps bulk scalar Poincaré representations to boundary Carrollian primary operators with weights $C$29. In the massless case, solving the intertwiner equations reproduces the known Mellin–Laplace basis,

$C$30

while the tachyonic and massive cases produce previously missing Carrollian bases. The tachyonic basis has real momentum support, whereas the massive basis necessarily involves a complex-support delta function and hence a complex momentum shift in scattering amplitudes. The same work emphasizes that Carrollian and celestial holography are not just changes of basis, because bulk and boundary states carry different conjugations and different natural time evolutions (Liu et al., 3 Jun 2026).

Taken together, these developments show that Carrollian conformal bases are not a single canonical object. They are a family of symmetry-adapted organizations of degenerate geometry, primary fields, boundary modes, scattering states, and dynamical generators. What remains stable across these realizations is the underlying conformal Carroll structure: a degenerate metric with a preferred null direction, an ultra-relativistic or null-boundary symmetry algebra with infinite extensions, and basis choices designed to make that structure manifest.